Searching for odderon in exclusive vector meson hadroproduction
aa r X i v : . [ h e p - ph ] A ug Searching for odderon in exclusive vector mesonhadroproduction
Leszek Motyka , ∗
1- II Institute for Theoretical Physics, University of Hamburg,Luruper Chaussee 149, D-22761, Germany2- Institute of Physics, Jagellonian UniversityReymonta 4, 30-059 Krak´ow, PolandIn this talk [1] estimates are presented of the odderon contributions to the exclusive
J/ψ and Υ production cross-sections at the Tevatron and the Large Hadron Collider. Theobtained cross-sections are compared to cross-sections of the dominant background sub-processes mediated by the photon exchange. Possible experimental cuts are proposedthat reduce the photon background.
Color neutral gluonic exchanges in high energy hadron scattering are naturally classified ac-cording to their C -parity. The C -even component is usually called the Pomeron. It gives thedominant contribution to vast majority of the measured cross-sections at high energies. The C -odd partner of the Pomeron is the odderon. In contrast to the Pomeron, the effects of theodderon are so weak that a compelling experimental evidence for the non-vanishing odderoncontribution has not been found yet. On the other hand, existence of the odderon exchangeis guaranteed within perturbative QCD. In the lowest non-trivial order, the QCD odderonis made of three gluons with the symmetric color structure. Beyond the leading order, theodderon amplitude can be described in the leading logarithmic ln s approximation (where √ s is the collisions energy) as a solution of the BKP evolution equation, a generalization ofthe famous BFKL equation for the Pomeron to a larger number of gluons. Thus, a successfulmeasurement of the QCD odderon exchange should provide some important insight into thehigh energy evolution of multi-gluon amplitudes in QCD.The main problem in odderon searches is the large background from the Pomeron which,if present, prohibits a measurement of the odderon contribution. In order to avoid thisproblem it is necessary to focus on processes in which, due to C -parity conservation, thePomeron contribution vanishes. This condition is fulfilled in exclusive production of mesonswith definite C -parity, at high energies. A number of such measurements was proposed andperformed at HERA, unfortunately with negative results.At hadron colliders, the Tevatron and the Large Hadron Collider (LHC) one expects toachieve an enhanced sensitivity to the odderon mediated processes, because of the strongcoupling of proton projectiles to gluons. One of the simplest processes that should besensitive to the odderon exchange in hadron collisions is the exclusive heavy vector mesonproduction [2], p ¯ p → p ¯ p + V or pp → pp + V where V = J/ψ,
Υ. In this talk [1] anattempt [3] to estimate the corresponding cross-sections for the Tevatron and the LHC isdescribed and prospects for the measurements are discussed. ∗ The support of the DFG grant no. SFB676 is gratefully acknowledged.
DIS 2008 pp kl lk k p kl kl l
21 2 1 2 p pp
211 2 ~ pp ~ a) b) c) d)Figure 1: The lowest order diagrams defining the Pomeron–odderon and Pomeron–photonfusion amplitudes of the vector meson production a) M P O , b) M O P , c) M γ P and d) M P γ . There are two main components of the exclusive vector meson hadroproduction amplitudes:one coming from the odderon-Pomeron fusion (Fig. 1a, 1b) and the other coming fromthe photon-Pomeron fusion (Fig. 1c, 1d). We estimated both contributions within the k T factorization approach, at the lowest order. In this approximation, the integrations overlongitudinal components of loop momenta can be performed exactly, and the non-trivialprocess dependent features are encoded in the impact factors Φ( . . . ), that describe couplingsof gluons to the scattering particles and depend only on the transverse momenta of thegluons, kkk i , lll i . As an example, we give here the amplitude of the diagram shown in Fig. 1a, M P O = − is ·
32! 3! 4(2 π ) R d lll lll d lll lll δ ( lll + lll − lll ) d kkk kkk d kkk kkk d kkk kkk δ ( kkk + kkk + kkk − kkk ) × δ ( kkk + lll ) kkk δ λ κ · Φ λ λ P ( lll , lll ) · Φ κ κ κ O ( kkk , kkk , kkk ) · Φ λ κ κ J/ψ ( lll , kkk , kkk ) . (1)Here kkk i , lll i are gluon transverse momenta and λ i , κ i are gluon color indices. Φ λ λ P andΦ κ κ κ O denote the impact factors of the proton, scattered via the Pomeron and odderonexchange respectively. Both the impact factors are obtained in the Fukugita-Kwieci´nskimodel of the proton, see [3] for details. The effective production vertex of the J/ψ mesonis denoted Φ λ κ κ J/ψ . It results from the perturbatively calculable fusion of three gluons into
J/ψ . In order to keep the notation of momenta lll i and kkk j most symmetric, we introduced anadditional, artificial vertex (denoted by the cross in Fig. 1) δ ( kkk + lll ) kkk δ λ κ connectingthe spectator gluons ( lll , λ ) and ( kkk , κ ). The ratio ·
32! 3! = is a combinatorial factor.The factors and correct the over-counting of diagrams introduced by factorizationin the scattering amplitudes of the impact factor with Pomeron and odderon exchanges,respectively. The factor 2 · lll i and kkk j . Diagrams shown in Fig. 1 give the lowest order amplitudes. A more realistic estimate ofthe production amplitudes requires taking into account the QCD evolution of the Pomeron.We represent this effect using a phenomenological enhancement factor E ( s, m V ), with V = DIS 2008 σ corr /dy J/ψ Υodderon photon odderon photonTevatron 0.3–1.3–5 nb 0.8–5–9 nb 0.7–4–15 pb 0.8–5–9 pbLHC 0.3–0.9–4 nb 2.4–15–27 nb 1.7–5–21 pb 5–31–55 pbTable 1: Estimates of cross-sections dσ corr /dy | y =0 given for the exclusive J/ψ and Υ pro-duction in pp and p ¯ p collisions by the odderon–Pomeron fusion and the photon-Pomeronfusion for the pessimistic–central–optimistic scenarios. J/ψ,
Υ. Besides that, it is necessary to take into account corrections coming from multiplescatterings, that may destroy the exclusive character of the process. Those effects will beexpressed as a gap survival factor S . An important model parameter that controls themagnitude of the proton impact factors is an effective strong coupling constant, ¯ α s . Thisparameter enters the Pomeron–odderon fusion cross-section in the fifth power. Thus, a cross-section, that takes into account necessary phenomenological improvements may be writtenas dσ corr /dy | y =0 = ¯ α s S E ( s, m V ) dσ/dy, where dσ/dy is the lowest order cross-sectionevaluated at ¯ α s = 1. We approximate the effects of QCD evolution of the Pomeron amplitudeby an exponential enhancement factor exp( λ ∆ y ) where ∆ y is the rapidity evolution length ofthe QCD Pomeron. Thus, for the central production one obtains E ( s, m V ) = ( x √ s/m V ) λ ,and the initial x -value for the gluon evolution is assumed to be x = 0 .
1. Following resultsfrom HERA, we take the effective Pomeron intercept λ = 0 . λ = 0 .
35) for the
J/ψ (Υ)production.The estimate of uncertainties introduced by ¯ α s and S is carried out together. Bydoing so, we follow the assumptions made in existing determinations of ¯ α s . For instance, alow value of ¯ α s ≃ . pp and p ¯ p scattering datain which S = 1 was taken. Analyzes of inclusive cross-sections and the exclusive vectormeson photoproduction yielded ¯ α s ≃ . −
1. Thus, we use S = 1 in our calculation ifthe low value of ¯ α s = 0 . S = 1 and ¯ α s = 0 . pessimistic scenario . In the optimistic scenario we use a ¯ α s = 1,combined with the gap survival factors obtained in the Durham two-channel eikonal model: S = 0 .
05 for the exclusive production at the Tevatron and S = 0 .
03 for the LHC. Thebest estimates should follow from the central scenario defined by ¯ α s = 0 . S = 0 . S = 0 .
03) at the Tevatron (LHC). Within the same model, we also analyze the Pomeron–photon contribution in a way analogous to the Pomeron–odderon contribution. The valuesof the phenomenologically improved cross-sections are summarized in Table 1 a . The photonand the odderon contributions do not interfere in the lowest order approximation and thecorresponding cross-sections may be treated independently. As seen from the table, thePomeron–odderon contributions are found to be uncertain, with a multiplicative uncertaintyfactor of 3–5. The ambiguities, however, cancel partially in the ratio of the Pomeron–odderoncontribution to the Pomeron–photon contribution evaluated in the same scenario. Thus,within the considered scenarios, the “odderon to photon ratio” R = [ dσ corr /dy ] / [ dσ corr γ /dy ]varies between 0.3 and 0.6 for J/ψ production at the Tevatron, and between about 0.06and 0.15 at the LHC. In the case of Υ, R varies between about 0.8 and 1.7 at the Tevatron a The photon cross-sections given in Table 1 are not meant to provide the most accurate estimate, butrather to asses the impact of model assumptions on the odderon/photon ratio.
DIS 2008 / s * d s / dp T [ / G e V ] p T2 [GeV ] a) J/ y odderon J/ y (photon) 0.01 0.1 1 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 / s * d s / dp T [ / G e V ] p T2 [GeV ] b) J/ y odderon J/ y (photon) Figure 2: The normalized differential cross-section ( dσ/dp T ) /σ for the exclusive production:a) p ¯ p → p ¯ p J/ψ , and b) pp → pp J/ψ as a function of meson p T for the odderon–Pomeronand the photon–Pomeron fusion.and between about 0.15 and 0.4 at the LHC. These numbers suggest that the odderoncontribution may well be of a similar magnitude to the photon contribution at the Tevatronand somewhat smaller than the photon contribution at the LHC.The obtained results clearly indicate that the photon exchange background to the odd-eron mediated processes is important in the p T -integrated cross-sections at central rapidities.Thus in the search for the odderon one should use also the available information on the trans-verse momentum distributions. In Fig. 2 normalized distributions of the meson p T are shownfor the odderon and photon contributions to the exclusive J/ψ production (the results for Υare similar). As seen from this figure, the relative importance of the odderon contributionincreases at larger meson p T , and the different p T -shape of the odderon and the photoncontributions may be used to perform an experimental separation between them. Anothermeasurement with an enhanced sensitivity to the odderon should be possible using forwardproton detectors. Assuming, that a forward detector at the LHC measures the proton A thatlost about x A ∼ .
01 of its energy in the pp → pp Υ process, one finds that the other proton, B , should lose a tiny fraction of about x B ∼ − of its energy. Although proton B cannotbe measured, the asymmetric kinematics implies that the proton B couples predominantlyto the Pomeron and proton A couples to the photon or the odderon. The photon exchange ischaracterized by a steep 1 /p T behavior and it leads to the p T distribution of proton A thatis concentrated at small momenta. The odderon induced p T -distribution is much broader.Thus, for instance, a cut of p T > . References [1] Slides: http://indico.cern.ch/materialDisplay.py?contribId=33&sessionId=16&materialId=slides&confId=24657 [2] A. Schafer, L. Mankiewicz and O. Nachtmann, Phys. Lett. B (1991) 419.[3] A. Bzdak, L. Motyka, L. Szymanowski and J. R. Cudell, Phys. Rev. D (2007) 094023.(2007) 094023.